A MATLAB differentiation matrix suite

ABSTRACT

A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.

REFERENCES

• 1: ABRAMOWITZ,M.AND STEGUN, I. E. 1964. Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC.
• 2: Fayez A. Alhargan, A complete method for the computations of Mathieu characteristic numbers of integer orders, SIAM Review, v.38 n.2, p.239-255, June 1996 [doi>10.1137/1038040]
• 3: BALTENSPERGER,R.AND BERRUT, J. P. 1999. The errors in calculating the pseudospectral differentiation matrices for Chebyshev-Gauss-Lobatto points. Comput. Math. Appl. 37,1, 41-48.
• 4: Alvin Bayliss , Andreas Class , Bernard J. Matkowsky, Roundoff error in computing derivatives using the Chebyshev differentiation matrix, Journal of Computational Physics, v.116 n.2, p.380-383, Feb. 1995 [doi>10.1006/jcph.1995.1036]
• 5: BERRUT, J. P. 1989. Barycentric formulae for cardinal (sinc-) interpolants. Numer. Math. 54, 703-718.
• 6: BOYD, J. P. 1984. The asymptotic coefficients of Hermite function series. J. Comput. Phys. 54, 382-410.
• 7: BOYD, J. P. 1989. Chebyshev and Fourier Spectral Methods. Springer-Verlag, Berlin, Germany.
• 8: Kenneth S. Breuer , Richard M. Everson, On the errors incurred calculating derivatives using Chebyshev polynomials, Journal of Computational Physics, v.99 n.1, p.56-67, March 1992 [doi>10.1016/0021-9991(92)90274-3]
• 9: CANUTO, C., HUSSAINI,M.Y.,QUARTERONI, A., AND ZANG, T. A. 1988. Spectral Methods in Fluid Dynamics. Springer-Verlag, Berlin, Germany.
• 10: COSTA,B.AND DON, W. S. 1999. Pseudopack 2000. See http://www.labma.ufrj.br/-bcosta/ PseudoPack2000/ Main.html.
• 11: Wai Sun Don , Alex Solomonoff, Accuracy and speed in computing the Chebyshev collocation derivative, SIAM Journal on Scientific Computing, v.16 n.6, p.1253-1268, Nov. 1995 [doi>10.1137/0916073]
• 12: DRAZIN,P.G.AND JOHNSON, R. S. 1989. Solitons: An Introduction. Cambridge University Press, New York, NY.
• 13: DRAZIN,P.G.AND REID, W. H. 1981. Hydrodynamic Stability. Cambridge University Press, New York, NY.
• 14: A. Dutt , M. Gu , V. Rokhlin, Fast algorithms for polynomial interpolation, integration, and differentiation, SIAM Journal on Numerical Analysis, v.33 n.5, p.1689-1711, Oct. 1996 [doi>10.1137/0733082]
• 15: Norman Eggert , Mary Jarratt , John Lund, Sinc function computation of the eigenvalues of Sturm-Liouville problems, Journal of Computational Physics, v.69 n.1, p.209-229, March 1, 1987 [doi>10.1016/0021-9991(87)90163-X]
• 16: FLUGGE, S. 1971. Practical Quantum Mechanics I. Springer-Verlag, Berlin, Germany.
• 17: FORNBERG, B. 1996. A Practical Guide to Pseudospectral Methods. Cambridge University Press, New York, NY.
• 18: FUNARO, D. 1992. Polynomial Approximation of Differential Equations. Springer-Verlag, Berlin, Germany.
• 19: FUNARO, D. 1993. Fortran routines for spectral methods. (available via anonymous FTP at ftp.ian.pv.cnr.it in pub/splib
• 20: GOTTLIEB, D., HUSSAINI,M.Y.,AND ORSZAG, S. A. 1984. Theory and applications of spectral methods. In Spectral Methods for Partial Differential Equations, R. Voigt, D. Gottlieb, and M. Hussaini, Eds. 1-54.
• 21: L. Greengard, Spectral integration and two-point boundary value problems, SIAM Journal on Numerical Analysis, v.28 n.4, p.1071-1080, Aug. 1991 [doi>10.1137/0728057]
• 22: GREENGARD,L.AND ROKHLIN, V. 1991. On the numerical solution of two-point boundary value problems. Comm. Pure Appl. Math. 44, 419-452.
• 23: Peter Henrici, Essentials of Numerical Analysis with Pocket Calculator Demonstrations, John Wiley & Sons, Inc., New York, NY, 1982
• 24: Peter Henrici, Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions, John Wiley & Sons, Inc., New York, NY, 1986
• 25: Weizhang Huang , David M. Sloan, The pseudospectral method for third-order differential equations, SIAM Journal on Numerical Analysis, v.29 n.6, p.1626-1647, Dec. 1992 [doi>10.1137/0729094]
• 26: HUANG,W.AND SLOAN, D. M. 1993. A new pseudospectral method with upwind features. IMA J. Num. Anal. 13, 413-430.
• 27: Weizhang Huang , David M. Sloan, The pseudospectral method for solving differential eigenvalue problems, Journal of Computational Physics, v.111 n.2, p.399-409, April 1994 [doi>10.1006/jcph.1994.1073]
• 28: ORSZAG, S. A. 1971. An accurate solution of the Orr-Sommerfeld equation. J. Fluid Mech. 50, 689-703.
• 29: PRYCE, J. D. 1993. Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis. Oxford University Press, Oxford, UK.
• 30: REDDY,S.C.,SCHMID,P.J.,BAGGETT,J.S.,AND HENNINGSON, D. S. 1998. On stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269-303.
• 31: SCHONFELDER, J. L. 1978. Chebyshev expansions for the error and related functions. Math. Comput. 32, 1232-1240.
• 32: SCHOOMBIE,S.W.AND BOTHA, J. F. 1981. Error estimates for the solution of the radial Schrodinger equation by the Rayleigh-Ritz finite element method. IMA J. Num. Anal. 1, 47-63.
• 33: Lawrence F. Shampine , Mark W. Reichelt, The MATLAB ODE Suite; , SIAM Journal on Scientific Computing, v.18 n.1, p.1-22, January. 1997 [doi>10.1137/S1064827594276424]
• 34: SHARMA, A. 1972. Some poised and nonpoised problems of interpolation. SIAM Rev. 14, 129-151.
• 35: STENGER, F. 1993. Numerical Methods Based on Sinc and Analytic Functions. Springer-Verlag, New York, NY.
• 36: Gilbert Strang, A proposal for toeplitz matrix calculations, Studies in Applied Mathematics, v.74 n.2, p.171-176, April 1986
• 37: Eitan Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM Journal on Numerical Analysis, v.23 n.1, p.1-10, Feb. 1986 [doi>10.1137/0723001]
• 38: Tao Tang, The Hermite spectral method for Gaussian-type functions, SIAM Journal on Scientific Computing, v.14 n.3, p.594-606, May 1993 [doi>10.1137/0914038]
• 39: Tao Tang , Manfred R. Trummer, Boundary Layer Resolving Pseudospectral Methods for Singular Perturbation Problems, SIAM Journal on Scientific Computing, v.17 n.2, p.430-438, March 1996 [doi>10.1137/S1064827592234120]
• 40: THE MATHWORKS,INC. 1998. MATLAB 5.2.
• 41: Lloyd N. Trefethen, Spectral methods in MatLab, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000
• 42: WALEFFE, F. 1995. Hydrodynamic stability and turbulence: Beyond transients to a selfsustaining process. Stud. Appl. Math. 95, 319-343.
• 43: WEIDEMAN, J. A. C. 1999. Spectral methods based on nonclassical orthogonal polynomials. In Approximations and Computation of Orthogonal Polynomials, W. Gautschi, G. Golub, and G. Opfer, Eds. Birkh~user, Basel, 239-251.
• 44: WEIDEMAN,J.A.C.AND TREFETHEN, L. N. 1988. The eigenvalues of second-order spectral differentiation matrices. SIAM J. Numer. Anal. 25, 1279-1298.
• 45: Bruno D. Welfert, Generation of Pseudospectral Differentiation Matrices I, SIAM Journal on Numerical Analysis, v.34 n.4, p.1640-1657, Aug. 1997 [doi>10.1137/S0036142993295545]
• 46: WIMP, J. 1984. Computation with Recurrence Relations. Pitman Publishing, Inc., Marshfield, MA.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE

      Nouns: MATLAB

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: Boundary value problems
      G.1.8 Partial Differential Equations
          Subjects: Spectral methods

General Terms:
Algorithms

Keywords:
MATLAB, differentiation matrices, pseudospectral methods, spectral collocation methods